Weakly stable irreducible Yang-Mills fields over $S^4$

TL;DR

This paper proves that weakly stable irreducible Yang-Mills fields on the 4-sphere are necessarily self-dual or anti-self-dual, confirming a special case of Yau's conjecture, and shows non-existence in certain algebraic conditions.

Contribution

It establishes a classification of weakly stable irreducible Yang-Mills fields on $S^4$ and rules out their existence under specific Lie algebra structures.

Findings

Weakly stable irreducible Yang-Mills fields are self-dual or anti-self-dual.

No irreducible Yang-Mills fields exist if the Lie algebra has a non-trivial abelian center.

The results address Yau's conjecture on $S^4$ for certain structure groups.

Abstract

Addressing Yau's conjecture (Problem 117) on $S^4$, we investigate the self-duality of weakly stable Yang-Mills fields under the assumption of irreducibility. For structure groups with a simple Lie algebra, we prove that any weakly stable irreducible connection must be either self-dual or anti-self-dual. Furthermore, we demonstrate that if the Lie algebra admits a non-trivial abelian center, no irreducible Yang-Mills fields can exist over $S^4$.